Stanisław Kasjan, Justyna Kosakowska
We prove the existence of Hall polynomials for $x^2$-bounded invariant subspaces of nilpotent linear operators.
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Two partial orders on Standard Young Tableaux with applications to invariant subspaces of nilpotent linear operators
该研究考察标准 Young 表上的两种偏序，并阐明其在幂零线性算子不变子空间理论中的应用。
Let $G$ be a semisimple algebraic group with Lie algebra $\mathfrak g$. For a nilpotent $G$-orbit $\mathcal O\subset\mathfrak g$, let $d_\mathcal O$ denote the maximal dimension of a subspace $V\subset \mathfrak g$ that is contained in the closure of $\mathcal O$. In this note, we prove that $d_\mathcal O \le \frac{1}{2}\dim\mathcal O$ and this upper bound is attained if and only if $\mathcal O$ is a Richardson orbit. Furthermore, if $V$ is $B$-stable and $\dim V= \frac{1}{2}\dim\mathcal O$, then $V$ is the nilradical of a polarisation of $\mathcal O$. Every nilpotent orbit closure has a disti…
We prove that the monoid of generic extensions of finite dimensional nilpotent $k[T]$-modules is isomorphic to the monoid of partitions (with addition of partitions). Moreover we give a combinatorial algorithm that calculates constant terms of classical Hall polynomials.
The main aim of the paper is to present a~combinatorial algorithm that, applying Littlewood-Richardson tableaux with entries equal to $1$, computes generic extensions of semisimple invariant subspaces of nilpotent linear operators. Moreover, we discuss geometric properties of generic extensions and their connections with combinatorics.
Let $k$ be a field. We consider triples $(V,U,T)$, where $V$ is a finite dimensional $k$-space, $U$ a subspace of $V$ and $T \:V \to V$ a linear operator with $T^n = 0$ for some $n$, and such that $T(U) \subseteq U$. Thus, $T$ is a nilpotent operator on $V$, and $U$ is an invariant subspace with respect to $T$. We will discuss the question whether it is possible to classify these triples. These triples $(V,U,T)$ are the objects of a category with the Krull-Remak-Schmidt property, thus it will be sufficient to deal with indecomposable triples. Obviously, the classification problem depends on $n…
We give a characterization of the finite groups having nilpotent or abelian Hall $π$-subgroups which can easily be verified from the character table.
The paper considers subspaces of the strictly upper triangular matrices, which are stable under Lie bracket with any upper triangular matrix. These subspaces are called ad-nilpotent ideals and there are Catalan number of such subspaces. Each ad-nilpotent ideal meets a unique largest nilpotent orbit in the Lie algebra of all matrices. The main result of the paper is that under an equivalence relation on ad-nilpotent ideals studied by Mizuno and others, the equivalence classes are the ad-nilpotent ideals with the same largest nilpotent orbit. We include two applications of the result, one to the…
We study geometric properties of varieties associated with invariant subspaces of nilpotent operators. There are reductive algebraic groups acting on these varieties. We give dimensions of orbits of these actions. Moreover, a combinatorial characterization of the partial order given by degenerations is described.
We define and investigate nilpotent Lie algebras associated with quadratic forms. We also present their connections with Lie algebras and Ringel-Hall algebras associated with representation directed algebras.
Let $\be$ be a Borel subalgebra of a complex simple Lie algebra $\g$. An ideal of $\be$ is called ad-nilpotent, if it is contained in $[\be,\be]$. The generators of an ad-nilpotent ideal give rise to an antichain in the poset of positive roots, and the whole theory can be expressed in a combinatorial fashion, in terms of antichains. The aim of this paper is to present a refinement of the enumerative theory of ad-nilpotent ideals for the case in which $\g$ has roots of different length. An antichain is called short, if it consists of short roots. We obtain, for short antichains, analogues of al…
In the present paper we study the derived Hall algebra for the bounded derived category of the nilpotent representations of a tame quiver over a finite field. We show that for any three given objects in the bounded derived category, the associated derived Hall numbers are given by a rational function in the cardinalities of ground fields.
The loop-nilpotent cohomological Hall algebra
为具有典范三次势的三倍箭图的循环幂零上同调霍尔代数建立洗牌代数模型，借此关联其与量子库仑分支代数，证明特化后的超交换性，给出两者生成元，刻画BPS李代数，得到Kac多项式新公式并证明相关猜想。
Commutativity of nilpotent cohomological Hall algebras of $\mathbf{A}^2$
研究$\mathbf{A}^2$的幂零上同调霍尔代数的交换性，通过结合对仿射BPS李代数上李括号的两个约束进行证明，还描述了特定作用下的等变幂零CoHAs，得到相关包络代数。
We determine the space of primary ideals in the group algebra $L^1(G)$ of a connected nilpotent Lie group by identifying for every $π\in\hat G $ the family ${\mathcal I}^π$ of primary ideals with hull $\{π\}$ with the family of invariant polynomials of a certain finite dimensional subspace ${\mathcal P}_Q^π$ of the space of polynomials ${\mathcal P}(G) $ on $G $.
We show a surprising link between singularity theory and the invariant subspace problem of nilpotent operators as recently studied by C. M. Ringel and M. Schmidmeier, a problem with a longstanding history going back to G. Birkhoff. The link is established via weighted projective lines and (stable) categories of vector bundles on those. The setup yields a new approach to attack the subspace problem. In particular, we deduce the main results of Ringel and Schmidmeier for nilpotency degree p from properties of the category of vector bundles on the weighted projective line of weight type (2,3,p), …
Primitive elements in the Hall algebra of a cyclic quiver
本文研究循环箭图幂零表示的Hall代数，给出了其中本原元素的显式公式，解决了该类代数中本原元的具体构造问题。
We determine the groups of automorphisms and their orbits for nilpotent Lie algebras of class 2 and small dimension, over arbitrary fields (including the characteristic 2 case).
We define and study the subspace of cuspidal functions for $G$-bundles on a class of nilpotent extensions $C$ of curves over a finite field. We show that this subspace is preserved by the action of a certain noncommutative Hecke algebra $\mathcal{H}_{G,C}$. In the case $G=\rm{GL}_2$, we construct a commutative subalgebra in $\mathcal{H}_{G,C}$ of Hecke operators associated with simple divisors. In the case of length 2 extensions and of $G=\rm{GL}_2$, we prove that the space of cuspidal functions (for bundles with a fixed determinant) is finite-dimensional and provide bounds on its dimension. I…
We show that the generic Hall algebra of nilpotent representations of an oriented cycle specialised at $q=0$ is isomorphic to the generic extension monoid in the sense of Reineke. This continues the work of Reineke.
We study the cohomological Hall algebra Y of a lagrangian substack of the moduli stack of representations of the preprojective algebra of an arbitrary quiver Q, and their actions on the cohomology of Nakajima quiver varieties. We prove that Y is pure and we compute its Poincare polynomials in terms of (nilpotent) Kac polynomials. We also provide a family of algebra generators. We conjecture that Y is equal, after a suitable extension of scalars, to the Yangian introduced by Maulik and Okounkov. As a corollary, we prove a variant of Okounkov's conjecture, which is a generalization of the Kac co…
This paper is a continuation of [8], in the direction of proving the conjecture that the spherical transform on a nilpotent Gelfand pair (N,K) establishes an isomorphism between the space of K-invariant Schwartz functions on N and the space of Schwartz functions restricted to the Gelfand spectrum properly embedded in a Euclidean space. We prove a result, of independent interest for the representation theoretical problems that are involved, which can be viewed as a generalised Hadamard lemma for K-invariant functions on N. The context is that of nilpotent Gelfand pairs satisfying Vinberg's cond…
Nilpotent cohomological Hall algebras of surfaces
本文利用上同调Hall代数理论，系统研究沿光滑曲面上固定曲线修改凝聚层的上同调Hecke算子，构造了具有集合支撑的凝聚层模栈，并定义了其幂零上同调Hall代数。
In his 1951 book "Infinite Abelian Groups", Kaplansky gives a combinatorial characterization of the isomorphism types of embeddings of a cyclic subgroup in a finite abelian group. In this paper we first use partial maps on Littlewood-Richardson tableaux to generalize this result to finite direct sums of such embeddings. We then focus on an application to invariant subspaces of nilpotent linear operators. We develop a criterion to decide if two irreducible components in the representation space are in the boundary partial order.
We prove that the classical $W$-algebra associated to a nilpotent orbit in a simple Lie-algebra can be constructed by preforming bihamiltonian, Drinfeld-Sokolov or Dirac reductions. We conclude that the classical $W$-algebra depends only on the nilpotent orbit but not on the choice of a good grading or an isotropic subspace. In addition, using this result we prove again that the transverse Poisson structure to a nilpotent orbit is polynomial and we better clarify the relation between classical and finite $W$-algebras.
Invariant Subspaces of Nilpotent Operators. Level, Mean, and Colevel: The Triangle $\Bbb T(n)$
本文研究带幂零算子的不变子空间范畴 $\mathcal S(n)$，提出用射影三角形 $\Bbb T(n)$ 编码层–共层比例，并刻画不可分解对象的支撑分布、对偶与 AR 平移几何及 $n=6$ 的 12 条中心直线。
Given partitions $α$, $β$, $γ$, the short exact sequences $0\to N_α\to N_β\to N_γ\to 0$ of nilpotent linear operators of Jordan types $α$, $β$, $γ$, respectively, define a constructible subset $\mathbb V_{α,γ}^β$ of an affine variety. Geometrically, the varieties $\mathbb V_{α,γ}^β$ are of particular interest as they occur naturally and since they typically consist of several irreducible components. In fact, each Littlewood-Richardson (LR-) tableau $Γ$ of shape $(α,β,γ)$ contributes one irreducible component $\overline{\mathbb V}_Γ$. We consider the partial order $Γ\leq_{\sf bound}^*\widetilde…
We study systems $(V,T,U_1,U_2)$ consisting of a finite dimensional vector space $V$, a nilpotent $k$-linear operator $T:V\to V$ and two $T$-invariant subspaces $U_1\subset U_2\subset V$. Let $\mathcal S(n)$ be the category of such systems where the operator $T$ acts with nilpotency index at most $n$. We determine the dimension types $(\dim U_1, \dim U_2/U_1, \dim V/U_2)$ of indecomposable systems in $\mathcal S(n)$ for $n\leq 4$. It turns out that in the case where $n=4$ there are infinitely many such triples $(x,y,z)$, they all lie in the cylinder given by $|x-y|,|y-z|,|z-x|\leq 4$. But not …
For a partition $β$, denote by $N_β$ the nilpotent linear operator of Jordan type $β$. Given partitions $β$, $γ$, we investigate the representation space ${}_2{\mathbb V}_γ^β$ of all short exact sequences $$ \mathcal E: 0\to N_α\to N_β\to N_γ\to 0$$ where $α$ is any partition with each part at most 2. Due to the condition on $α$, the isomorphism type of a sequence $\mathcal E$ is given by an arc diagram $Δ$; denote by ${\mathbb V}_Δ$ the subset of ${}_2{\mathbb V}_γ^β$ of all sequences isomorphic to $\mathcal E$. Thus, the space ${}_2{\mathbb V}_γ^β$ carries a stratification given by the subse…
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